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Theorem inindif 30205
Description: See inundif 4423. (Contributed by Thierry Arnoux, 13-Sep-2017.)
Assertion
Ref Expression
inindif ((𝐴𝐶) ∩ (𝐴𝐶)) = ∅

Proof of Theorem inindif
StepHypRef Expression
1 inss2 4203 . . . 4 (𝐴𝐶) ⊆ 𝐶
21orci 859 . . 3 ((𝐴𝐶) ⊆ 𝐶𝐴𝐶)
3 inss 4212 . . 3 (((𝐴𝐶) ⊆ 𝐶𝐴𝐶) → ((𝐴𝐶) ∩ 𝐴) ⊆ 𝐶)
42, 3ax-mp 5 . 2 ((𝐴𝐶) ∩ 𝐴) ⊆ 𝐶
5 inssdif0 4326 . 2 (((𝐴𝐶) ∩ 𝐴) ⊆ 𝐶 ↔ ((𝐴𝐶) ∩ (𝐴𝐶)) = ∅)
64, 5mpbi 231 1 ((𝐴𝐶) ∩ (𝐴𝐶)) = ∅
Colors of variables: wff setvar class
Syntax hints:  wo 841   = wceq 1528  cdif 3930  cin 3932  wss 3933  c0 4288
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1787  ax-4 1801  ax-5 1902  ax-6 1961  ax-7 2006  ax-8 2107  ax-9 2115  ax-10 2136  ax-11 2151  ax-12 2167  ax-ext 2790
This theorem depends on definitions:  df-bi 208  df-an 397  df-or 842  df-tru 1531  df-ex 1772  df-nf 1776  df-sb 2061  df-clab 2797  df-cleq 2811  df-clel 2890  df-nfc 2960  df-rab 3144  df-v 3494  df-dif 3936  df-in 3940  df-ss 3949  df-nul 4289
This theorem is referenced by:  resf1o  30392  gsummptres  30617  indsumin  31180  measunl  31374  carsgclctun  31478  probdif  31577  hgt750lemd  31818
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